Regular polytopes of two or more dimensions in Euclidean space have been extensively studied. The regular polygons and polyhedra, or 'Platonic Solids', have been known since ancient times and the polytopes of four and more dimensions were discovered nearly a century ago by Schläfli. These figures commend themselves to our attention not only on account of their aesthetic appeal but also because their symmetries form important groups isomorphic to groups of real orthogonal matrices.

The real orthogonal matrix may be considered as a special case of the more general unitary matrix with complex coefficients which satisfies the relation $\bar{T}' = T^{-1}$ where the dash denotes transposition and the bar the complex conjugate. Many groups of unitary matrices are known and it is natural to inquire whether these can be exhibited as symmetry groups of 'complex polytopes' constructed in 'unitary space' ...

Unitary groups are of particular importance since it can be shown that all finite collineation groups can be reduced to this form.

The objects of this paper are:

To introduce, for the first time, the concept of a complex polytope.
To develop the theory of complex polytopes as far as a complete enumeration of the regular polytopes.

I am very much indebted to Dr J A Todd for his help and advice in the preparation of this paper.

I must express my indebtedness to J A Todd and H S M Coxeter for their advice and suggestions in carrying out the investigations described in this paper. I am especially grateful to the former for undertaking the formidable task of checking the abstract definitions.

The dissertation consists of a description of a novel method of investigating certain well-known collineation groups in projective space, by the use of 'complex polytopes'. Real polytopes, the analogues in space of any number of dimensions of polygons and polyhedra, have been very fully investigated.

Leaving a Readership at University College London, in 1954 Ambrose Rogers went to Birmingham as Mason Professor of Pure Mathematics. In collaboration with Geoffrey Shephard and James Taylor during that period his interest in convex geometry and Hausdorff Measure Theory widened. In particular, with Geoffrey Shephard, he produced sharp bounds for the volume of a difference body, a problem which had been open for 30 years.

Shephard's pamphlet was sold at the Porters' Lodge and, with the growth of tourism, became widely known as the authority on the dial.

This book is based on a course of lectures on linear algebra given to second-year Honours students in the University of Birmingham.

Branko met Geoffrey Shephard in 1975. They decided to write a book on Visual Geometry. This was a huge undertaking. So they decided to start with tilings and patterns as a first step in their program. After 11 years of research, tracing ancient and current places where tilings were used, the book was published.

This paper represents, we believe, an excellent example of international cooperation. The four authors, from four different countries, have never met and do not have a common language. The preparation of the final manuscript and diagrams was undertaken by two of the authors (B G and G C S) who take full responsibility for any errors that may occur.

"Remarkable ... It will surely remain the unique reference in this area for many years to come," Roger Penrose, Nature; "... an outstanding achievement in mathematical education," Bulletin of the London Mathematical Society; "I am enormously impressed ... Will be the definitive reference on tiling theory for many decades. Not only does the book bring together older results that have not been brought together before, but it contains a wealth of new material ... I know of no comparable book," Martin Gardner.

What Grünbaum and Shephard have done, in a dazzling display of scholarship, erudition, and research, is collect in one volume a compendium of the accumulated knowledge about tilings and patterns developed by a wide range of individuals including artisans and craftsmen, mathematicians, crystallographers, and physicists. In doing so they were forced to take a fresh look at what was known, what was thought to be known, and what had yet to be investigated, and to provide a framework in which all of this information could be succinctly put down. The project, which was started about 10 years ago and has only now been brought to (partial) completion, is well worth the wait.

This carefully composed book summarises and extends some of the most important parts of B Grünbaum's treatise 'Convex polytopes' (1967). It is a more readable exposition, and brings the subject up to date.

It seems to us that geometry of this kind described here (concerning polyhedra, space-fillings, symmetry groups, etc.) is both interesting and exciting. It is a tragedy that our present educational system completely ignores these subjects and the little time available for teaching geometry seems to be devoted to comparatively simple and uninteresting topics. Perhaps this is one of the reasons why there still remain so many unsolved geometrical problems in the familiar three-dimensional space in which we live.

[Geoffrey Shephard was] an inspiring speaker and with a sense of theatre and an ability to hold an audience - at least in the few University of East Anglia lectures of his which I attended. He reminded one of G H Hardy's dictum along the lines of: You may have complicated things to say, but you should always say them simply. It comes as no surprise that he provided a prize for work in mathematics which can be explained to people who have no prior knowledge in that field.

Following a very generous donation made by Professor Geoffrey Shephard, the London Mathematical Society will, in 2015, introduce a new prize. The prize, to be known as the Shephard Prize will be awarded biennially. The award will be made to a mathematician (or mathematicians) based in the UK in recognition of a specific contribution to mathematics with a strong intuitive component which can be explained to those with little or no knowledge of university mathematics, though the work itself may involve more advanced ideas.
...
Professor Shephard himself is Professor of Mathematics at the University of East Anglia whose main fields of interest are in convex geometry and tessellations. Professor Shephard is one of the longest-standing members of the London Mathematical Society, having given more than sixty years of membership. The Society wishes to place on record its thanks for his support in the establishment of the new prize.

He continued to be a regular visitor to the School of Mathematics since that time [1987]; it is only in the last few years that these visits became less frequent.